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Computing the Pfaffian of a skew-symmetric matrix is a problem that arises invarious fields of physics. Both computing the Pfaffian and a related problem,computing the canonical form of a skew-symmetric matrix under unitarycongruence, can be solved easily once the skew-symmetric matrix has beenreduced to skew-symmetric tridiagonal form. We develop efficient numericalmethods for computing this tridiagonal form based on Gauss transformations,using a skew-symmetric, blocked form of the Parlett-Reid algorithm, or based onunitary transformations, using block Householder transformations and Givensrotations, that are applicable to dense and banded matrices, respectively. Wealso give a complete and fully optimized implementation of these algorithms inFortran, and also provide Python, Matlab and Mathematica implementations forconvenience. Finally, we apply these methods to compute the topological chargeof a class D nanowire, and show numerically the equivalence of definitionsbased on the Hamiltonian and the scattering matrix.

Efficient numerical computation of the Pfaffian for dense and banded skew-symmetric matrices